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Flexible two-point selection approach for characteristic function-based parameter estimation of stable laws.

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Infinite ergodicity that preserves the Lebesgue measure.

Ken-Ichi Okubo¹, Ken Umeno¹

¹Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan.

Chaos (Woodbury, N.Y.)

|April 3, 2021

Summary

This study proves infinite one-dimensional dynamical systems preserve Lebesgue measure and are ergodic. These systems exhibit weak chaos and their Lyapunov exponent distributions follow the Mittag-Leffler distribution.

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Area of Science:

Dynamical Systems Theory
Ergodic Theory
Measure Theory

Background:

Dynamical systems can exhibit complex behaviors, including exactness and divergence of orbits.
Understanding critical parameter points where weak chaos emerges is crucial.
Previous work by Adler and Weiss laid groundwork for analyzing such systems.

Purpose of the Study:

To prove that a countably infinite set of one-parameterized one-dimensional dynamical systems preserve Lebesgue measure.
To demonstrate the ergodicity of these systems with respect to the Lebesgue measure.
To investigate the behavior of these systems at critical parameter points associated with weak chaos.

Main Methods:

Analytical proof of measure preservation and ergodicity for a class of one-dimensional dynamical systems.
Connecting parameter regions of exactness and divergent orbits.
Numerical simulations to analyze the distribution of the normalized Lyapunov exponent.

Main Results:

A countably infinite number of one-parameterized one-dimensional dynamical systems are shown to preserve the Lebesgue measure and be ergodic.
These systems bridge the gap between exact dynamical systems and those with diverging orbits.
The normalized Lyapunov exponent distributions for these systems were found to follow the Mittag-Leffler distribution of order 1/2.

Conclusions:

The study establishes a significant class of measure-preserving and ergodic dynamical systems.
The findings generalize and extend previous results in ergodic theory.
The emergence of the Mittag-Leffler distribution highlights a novel statistical property in these weakly chaotic systems.